To find the area of the quadrilateral with the given four vertices, we may use the formula given below.

**Question 1 :**

Let P(11,7) , Q(13.5, 4) and R(9.5, 4) be the mid- points of the sides AB, BC and AC respectively of triangle ABC . Find the coordinates of the vertices A, B and C. Hence find the area of triangle ABC and compare this with area of triangle PQR

**Solution :**

To find the vertices of the triangle from the midpoint of the sides, please visit the page "https://www.onlinemath4all.com/how-to-find-the-vertices-of-a-triangle-if-the-midpoints-are-given.html"

Vertex A :

= (11 + 9.5 - 13.5, 7 + 4 - 4)

= A (7, 7)

Vertex B :

= (11 + 13.5 - 9.5, 7 + 4 - 4)

= B (15, 7)

Vertex C :

= (13.5 + 9.5 - 11, 4 + 4 - 7)

= C (12, 1)

**Area of triangle ABC :**

= (1/2)[(49 + 15 + 84) - (105 + 84 + 7)]

= (1/2)[148 - 196]

= 48/2

Area of triangle ABC = 24 square units

**Area of triangle PQR :**

**P(11,7) , Q(13.5, 4) and R(9.5, 4) **

= (1/2)[(44 + 54 + 66.5) - (94.5 + 38 + 44)]

= (1/2)[164.5 - 176.5]

= (1/2) (12)

Area of triangle PQR = 6

Area of triangle ABC = 4 (Area of triangle PQR)

Let us look into the next example on "Word Problems to Find Area of Quadrilaterals with Vertices".

**Question 2 :**

In the figure, the quadrilateral swimming pool shown is surrounded by concrete patio. Find the area of the patio.

**Solution :**

To find the area of patio, we have to subtract area of EFGH from area of ABCD.

**Area of ABCD :**

= (1/2)[(16 + 80 + 36 + 80) - (-64 - 24 - 100 - 24)]

= (1/2)[(212)-(-212)]

= (1/2)[212+212]

= 212 square units

**Area of EFGH :**

= (1/2)[(6 + 42 + 12 + 30) - (-30 - 6 - 42 - 12)]

= (1/2)[(90)-(-90)]

= (1/2)[90+90]

Area of EFGH = 90 square units

Area of patio = 212 - 90

= 122 square units.

**Question 3 :**

A triangular shaped glass with vertices at A(-5,-4) , B(1,6) and C(7,-4) has to be painted. If one bucket of paint covers 6 square feets, how many buckets of paint will be required to paint the whole glass, if only one coat of paint is applied.

**Solution :**

Area of triangle ABC =

= (1/2)[(-30 - 4 - 28) - (-4 + 42 + 20)]

= (1/2)[-62 - (58)]

= (1/2)[-62 - 58]

= (1/2)(120)

= 60 square feet

Area covered by one bucket of paint = 6 square feets

Required number of bucket = 60 / 6

= 10 buckets

**Question 4 :**

In the figure, find the area of (i) triangle AGF (ii) triangle FED (iii) quadrilateral BCEG

**Solution :**

(i) triangle AGF

= (1/2) [(-2.5 - 13.5 - 6) - (-13.5 - 1 - 15)]

= (1/2) [(-22) - (-29.5)]

= (1/2) (-22+29.5)

= (1/2)[7.5]

= 3.75 square units.

(ii) triangle FED

F (-2, 3) E (1.5, 1) and D (1, 3)

= (1/2)[(-2 + 4.5 + 3) - (4.5 + 1 - 6)]

= (1/2)[(5.5) - (-0.5)]

= (1/2)6

Area of triangle FED = 3 square units.

(iii) B (-4, -2) C (2, -1) E (1.5, 1) G (-4.5, 0.5)

= (1/2)[(4 + 2 + 0.75 + 9) - (-4 - 1.5 - 4.5 - 2)]

= (1/2)[(15.75) - (-12)]

= (1/2)(15.75 + 12)

= (1/2)(27.75)

= 13.875 square units.

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